Diophantine inequality involving binary forms

Let r = 2d−1 + 1. We investigate the diophantine inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| {\sum\limits_{i = 1}^r {\lambda _i \Phi _i \left( {x_i ,y_i } \right) + \eta } } \right| < \left( {\mathop {\max }\limits_{1 \leqslant i \leqslant r} \left\{ {\left| {x_i } \right|,\left| {y_i } \right|} \right\}} \right)^{ - \sigma } ,$$\end{document} where Φi(x, y) ∈ ℤ[x, y] (1 ⩽ i ⩽ r) are nondegenerate forms of degree d = 3 or 4.